Call a biased coin a p-coin if it comes up heads with probability p and tails with probability 1-p. We say that a p-coin simulates a q-coin if by flipping a p-coin repeatedly (some fixed finite number of times) one can simulate the behavior of a q-coin.

For example, a fair coin can be used to simulate a 3/4-coin by using two flips and defining a pseudo-head to be any two-flip sequence with at least one real head. The chance of a pseudo-head coming up is 3/4, so we have simulated a 3/4-coin.

1. Find a rational value p such that a p-coin can simulate both a 1/2-coin and a 1/3-coin, or prove that no such value exists.

2. Find an irrational value p such that a p-coin can simulate both a 1/2-coin and a 1/3-coin, or prove that no such value exists.

For instance, a 1/sqrt(2) -coin can simulate a 1/2 coin, in exactly the same way that a 1/2 coin can simulate a 3/4 coin. 

(Incidentally, the 3/4 coin can be simulated by a rational-coin because the denominator of 3/4 is a square.  Perhaps a non-prime is sufficient.)

Posted by Steve Herman on 2005-02-19 10:53:43